Atom and Bond Fukui Functions and Matrices: A Hirshfeld‐I Atoms‐in‐Molecule Approach

The Fukui function is often used in its atom‐condensed form by isolating it from the molecular Fukui function using a chosen weight function for the atom in the molecule. Recently, Fukui functions and matrices for both atoms and bonds separately were introduced for semiempirical and ab initio levels of theory using Hückel and Mulliken atoms‐in‐molecule models. In this work, a double partitioning method of the Fukui matrix is proposed within the Hirshfeld‐I atoms‐in‐molecule framework. Diagonalizing the resulting atomic and bond matrices gives eigenvalues and eigenvectors (Fukui orbitals) describing the reactivity of atoms and bonds. The Fukui function is the diagonal element of the Fukui matrix and may be resolved in atom and bond contributions. The extra information contained in the atom and bond resolution of the Fukui matrices and functions is highlighted. The effect of the choice of weight function arising from the Hirshfeld‐I approach to obtain atom‐ and bond‐condensed Fukui functions is studied. A comparison of the results with those generated by using the Mulliken atoms‐in‐molecule approach shows low correlation between the two partitioning schemes.     

Some basic properties of the correlation matrices

On the basis of the properties of correlation matrices, it is shown here that the set of all the first‐order transition reduced density matrices of a system provide complete information about that system. Also, the interrelation between the properties of the correlation matrix and the 2‐RDM N‐representability conditions is studied. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Bond fukui indices: Comparison of frozen molecular orbital and finite differences through mulliken populations

Bond Fukui functions and matrices are introduced for ab initio levels of theory using a Mulliken atoms in molecules model. It is shown how these indices may be obtained from first‐order density matrix derivatives without need for going to second‐order density matrices as in a previous work. The importance of taking into account the nonorthogonality of the basis in ab initio calculations is shown, contrasting the present results with previous work based on Hückel theory. It is shown how the extension of Fukui functions to Fukui matrices allows getting more insight into the nature of bond Fukui functions. All presently introduced indices respect the necessary normalization conditions and include the classical single atom condensed Fukui functions. © 2013 Wiley Periodicals, Inc.     

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.     

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     

One‐particle density matrix functional for correlation in molecular systems

Based on the analysis of the general properties for the one‐ and two‐particle reduced density matrices, a new natural orbital functional is obtained. It is shown that by partitioning the two‐particle reduced density matrix in an antisymmeterized product of one‐particle reduced density matrices and a correction Γ^c we can derive a corrected Hartree–Fock theory. The spin structure of the correction term from the improved Bardeen–Cooper–Schrieffer theory is considered to take into account the correlation between pairs of electrons with antiparallel spins. The analysis affords a nonidempotent condition for the one‐particle reduced density matrix. Test calculations of the correlation energy and the dipole moment of several molecules in the ground state demonstrate the reliability of the formalism. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 94: 317–323, 2003     

Closed-form expressions for correlated density matrices: application to dispersive interactions and example of He2

Empirically correlated density matrices of N-electron systems are investigated. Closed-form expressions are derived for the one- and two-electron reduced density matrices from a pairwise correlated wave function. Approximate expressions are then proposed which reflect dispersive interactions between closed-shell centrosymmetric subsystems. Said expressions clearly illustrate the consequences of second-order correlation effects on the reduced density matrices. Application is made to a simple example: the He(2) system. Reduced density matrices are explicitly calculated, correct to second order in correlation, and compared with approximations of independent electrons and independent electron pairs. The models proposed allow for variational calculations of interaction energies and equilibrium distance as well as a clear interpretation of dispersive effects on electron distributions. Both exchange and second order correlation effects are shown to play a critical role on the quality of the results.     

Quantum Theory of ICN Photodissociation: Density Matrices of Photofragments from a Parallel Transition

A quantum treatment on ICN photodissociation from an initial parallel transition (Ω′ = 0 ← Ω″ = 0) to 1 the asymptote CN(|)+ I(²P_1/2) is presented. Density matrices of both photofragments are derived, and explicit expressions of the state multipoles in terms of the angular momentum coupling coefficients and the rotation‐bending factors have been obtained. The present theoretical framework provides a foundation to study photofragments with non‐null electronic and/or spin angular momenta. To investigate the angular momentum polarizations phenomena, these density matrices can play a prime role in laser‐based detections of state‐selected photofragments.     

The Fukui matrix: a simple approach to the analysis of the Fukui function and its positive character

The Fukui matrix is introduced as the derivative of the one-electron reduced density matrix with respect to a change in the number of electrons under constant external potential. The Fukui matrix extends the Fukui function concept: the diagonal of the Fukui matrix is the Fukui function. Diagonalizing the Fukui matrix gives a set of eigenvectors, the Fukui orbitals, and accompanying eigenvalues. At the level of theory used, there is always one dominant eigenvector, with an eigenvalue equal to 1. The remaining eigenvalues are either zero or come in pairs with eigenvalues of the same magnitude but opposite sign. Analysis of the frontier molecular orbital coefficient in the eigenvector with eigenvalue 1 gives information on the quality of the frontier molecular orbital picture. The occurrence of negative Fukui functions can be easily interpreted in terms of the nodal character of the dominant eigenvector versus the characteristics of the remaining eigenvectors and eigenvalues.     

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.     

A new partitioning scheme for molecular interacting systems within a multiconfigurational or monoconfigurational Hartree–Fock formalism

A new method, based on the spatial decomposition of the reduced‐density and pair‐density matrices and the indistinguishable integrals formalism, is introduced to partition the molecular and stabilization energies into meaningful fragments. These are defined as entirely flexible variable‐size entities, for example, atoms, group of atoms, ions, and interacting monomers. This new partitioning scheme is especially appropriated to study systems in which a directly bonded group‐transfer process occurs. In these cases, the stabilization energies are partitioned into an intrafragment component, associated with the difference of intrinsic affinity to the transferred group between the involved fragments, and an interfragment component, associated with the difference of the magnitude of the interaction between the fragments in the initial and final binding complexes. This method was applied to the study of the arginine–carboxylate interactions, allowing us to have insight into what really happens in this system. Two (zwitterionic and neutral) binding complexes can be considered. The main effects accountable for the preferential stabilizations of the binding complexes are determined to be basis‐set independent. The zwitterionic complex is favored by the interfragment component, while the neutral complex is favored by the larger intrinsic proton affinity of the acetate relatively to the methylguanidium. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 157–176, 1999     

Atomic charges,dipole moments,and Fukui functions using the Hirshfeld partitioning of the electron density

In the Hirshfeld partitioning of the electron density, the molecular electron density is decomposed in atomic contributions, proportional to the weight of the isolated atom density in the promolecule density, constructed by superimposing the isolated atom electron densities placed on the positions the atoms have in the molecule. A maximal conservation of the information of the isolated atoms in the atoms-in-molecules is thereby secured. Atomic charges, atomic dipole moments, and Fukui functions resulting from the Hirshfeld partitioning of the electron density are computed for a large series of molecules. In a representative set of organic and hypervalent molecules, they are compared with other commonly used population analysis methods. The expected bond polarities are recovered, but the charges are much smaller compared to other methods. Condensed Fukui functions for a large number of molecules, undergoing an electrophilic or a nucleophilic attack, are computed and compared with the HOMO and LUMO densities, integrated over the Hirshfeld atoms in molecules.     

Communication: Constrained search formulation of the ground state energy as a functional of an idempotent one-matrix

Despite the fact that idempotent one-particle reduced density matrices are pervasive in quantum chemistry, the understanding of a general energy functional of such idempotent density matrices for the ground state energy has been lacking. By a constrained search, we show the structure of the general functional, illuminating the contributions from various terms. For the examples of the "best idempotent density matrix" and Kohn-Sham idempotent density matrices, we contrast the functional forms and suggest how the best idempotent density matrix approach may be a good starting point for further development.     

Theoretical studies on geometry,solvent effect,and photochromic mechanism of two bis‐heterocyclic compounds containing pyrazolone ring

Two bis‐heterocyclic compounds containing pyrazolone ring, 1‐phenyl‐3‐methyl‐4‐(6‐hydro‐4‐amino‐5‐sulfo‐2,3‐pyrazine)‐pyrazole‐5‐one and 1‐phenyl‐3‐methyl‐4‐(6‐hydro‐4‐methylamino‐5‐sulfo‐2,3‐pyrazine)‐pyrazole‐5‐one, are investigated to gain a deeper insight into their geometries and photochromic mechanism by applying density functional theory. The solvent effects are simulated using the polarizable continuum model of the self‐consistent reaction field theory. Bader's atom‐in‐molecule theory is used to investigate the nature of hydrogen bonds. The data of energy, dipole moments, and the condensed Fukui functions have been calculated to assess the stability and reactivity of the title compounds. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Performance of density functional theory methods to describe intramolecular hydrogen shifts

The performance of three exchange and correlation density functionals, LDA, BLYP and B3LYP, with four basis sets is tested in three intramolecular hydrogen shift reactions. The best reaction and activation energies come from the hybrid functional B3LYP with triple-ζ basis sets, when they are compared with high-level post-Hartree-Fock results from the literature. For a fixed molecular geometry, the electrophilic Fukui function is computed from a finite difference approximation. Fukui function shows a small dependence with both the exchange and correlation functional and the basis set. Evolution of the Fukui function along the reaction path describes important changes in the basic sites of the corresponding molecules. These results are in agreement with the chemical behavior of those species.     

Electron density, exchange-correlation density, and bond characterization from the perspective of the valence-bond theory. I. Two simple analytical cases

In this work, using a valence-bond wave function we obtain analytical expressions for the first- and second-order reduced density matrices of two simple, but quite representative, cases of diatomic molecular systems, namely, H2 and LiH. A detailed study of their exchange-correlation density is performed for both equilibrium and nonequilibrium internuclear distances, discriminating the parallel- and antiparallel-spin contributions. The results show that the behavior of the exchange-correlation density clearly changes with the character of the bond, making it possible to obtain a good deal of information regarding the type of the bond interaction.     

Extraction of fullerenes from environmental matrices as affected by solvent characteristics and analyte concentration

Fullerenes possess unique chemical properties that make the isolation of these compounds from heterogeneous environmental matrices difficult. For example, previous reports indicate that toluene‐based extraction techniques vary in their ability to extract C₆₀, especially from highly carbonaceous solid matrices. Here, we examined the effects of (i) solvent type (toluene alone versus an 80:20 v/v mixture of toluene and 1‐methylnaphthalene) and (ii) analyte concentration on the extraction efficiency of an isotopically labeled surrogate compound, ¹³C₆₀. The toluene/1‐methylnaphthalene mixture increased fullerene extraction efficiency from carbon lampblack by a factor of five, but was not significantly different from 100% toluene when applied to wood stove soot or montmorillonite. Recovery of the ¹³C₆₀ surrogate declined with decreasing analyte concentration. The usefulness of isotopically labeled surrogate is demonstrated and the study provides a quantitative assessment regarding the dependence of fullerene extraction efficiencies on the geochemical characteristics of solid matrices.     

Nonlinear optical limiters of pulsed laser radiation based on carbon‐containing nanostructures in viscous and solid matrices

Results of the study of optical limiters of pulsed laser radiation based on nonlinear effects in carbon nanostructures placed into viscous and solid matrices are presented. A nonlinear optical limiting was studied by nanomaterials based on multi‐wall polyhedral carbon nanostructures (astralens) placed in a sol–gel matrix. Similar studies for single‐wall and multi‐wall carbon‐containing nanotubes placed in polymer matrices with various viscosities were performed. No additional mechanism of optical limiting due to electron structure of single‐wall carbon‐containing nanotubes at their introduction into viscous and solid composite media was found. An influence of polymer matrix composition containing carbon nanotubes (CNTs) on a threshold and ratio of attenuation of laser radiation was demonstrated. The best limiting characteristics were observed at placing CNT into polymethylsiloxane matrix. An effect of “self‐healing” of a medium after laser radiation passage through high viscous liquids was obtained. The high parameters of nonlinear optical limiting (the threshold of limiting 10^?5 J, ratio of attenuation 10³) achieved for the composite material CNT (HiPCO High‐Pressure Carbon Monoxide) and carbon nanofibers in high viscous and solid polymethylsiloxane media allow the design of protective filters for laser radiation operating in wide spectral range. Copyright © 2014 John Wiley & Sons, Ltd.     

Hierarchy equations for reduced density matrices in different representations

We derive exact relationships for the reduced density matrices in representations where the transformation matrix is a product of one-body transformation matrices. We specialize to the momentum and onebody energy representations. By decoupling the equations we are able to write the Hartree-Fock equation in terms of the first-order density matrix in an arbitrary representation. Applications to reduced local energy and the correlation problem are discussed.     

Comparison among four different ways to condense the Fukui function

Four different ways to condense the Fukui function are compared. Three of them perform a numerical integration over different basins to define the condensed Fukui function, and the other one is the most traditional Fukui function using Mulliken population analysis. The basins are chosen to be the basins of the electron density (AIM), the basins of the electron localization function (ELF), and the basins of the Fukui function itself. The use of the last two basins is new and presented for the first time here. It is found that the last three methods yield results which are stable against a change in the basis set. The condensed Fukui function using the basins of the ELF is not able to give information on the reactivity of an acceptor molecule. In general, the condensed Fukui function using the basins of the density or the basins of the Fukui function describe the reactivity trends well. The latter is preferred, because it only contains information about the Fukui function itself and it gives the right information for donor as well as acceptor centers.