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The energy of a many-particle system is not convex with respect to particle number for r−k interparticle repulsion potentials if k > log34 ≈ 1.262. With such potentials, some finite electronic systems have ionization potentials that are less than the electron affinity: they have negative band gap (chemical hardness). Although the energy may be a convex function of the number of electrons (for which k = 1), it suggests that finding an analytic proof of convexity will be very difficult. The bound on k is postulated to be tight. An apparent signature of non-convex behavior is that the Dyson orbital corresponding to the lowest-energy mode of electron attachment has a vanishingly small amplitude. [ABSTRACT FROM AUTHOR]

An overview of mathematical properties of the non-local second order derivatives of the canonical, grand canonical, isomorphic, and grand isomorphic ensembles is given. The significance of their positive or negative semidefiniteness and the implications of these properties for atoms and molecules are discussed. Based on this property, many other interesting properties can be derived, such as the expansion in eigenfunctions, bounds on the diagonal and off-diagonal elements, and the eigenvalues of these kernels. We also prove Kato's theorem for the softness kernel and linear response and the dissociation limit of the linear responses as the sum of the linear responses of the individual fragments when dissociating a system into two non-interacting molecular fragments. Finally, strategies for the practical calculation of these kernels, their eigenfunctions, and their eigenvalues are discussed. [ABSTRACT FROM AUTHOR]

We present how the framework of the temperature-dependent chemical reactivity theory can describe the panorama of different types of interactions between an electronic system and external reagents. The key reactivity indicators are responses of an appropriate state function (like the energy or grand potential) to the variables that determine the state of the system (like the number of electrons/chemical potential, external potential, and temperature).We also consider the response of the average electron density to appropriate perturbations. We present computable formulas for these reactivity indicators and discuss their chemical utility for describing electronic, electrostatic, and thermal changes associated with chemical processes. [ABSTRACT FROM AUTHOR]

We explore the local and nonlocal response functions of the grand canonical potential density functional at nonzero temperature. In analogy to the zero-temperature treatment, local (e.g., the average electron density and the local softness) and nonlocal (e.g., the softness kernel) intrinsic response functions are defined as partial derivatives of the grand canonical potential with respect to its thermodynamic variables (i.e., the chemical potential of the electron reservoir and the external potential generated by the atomic nuclei). To define the local and nonlocal response functions of the electron density (e.g., the Fukui function, the linear density response function, and the dual descriptor), we differentiate with respect to the average electron number and the external potential. The well-known mathematical relationships between the intrinsic response functions and the electron-density responses are generalized to nonzero temperature, and we prove that in the zero-temperature limit, our results recover well-known identities from the density functional theory of chemical reactivity. Specific working equations and numerical results are provided for the 3-state ensemble model. [ABSTRACT FROM AUTHOR]

We describe a practical algorithm for constructing the Kohn-Sham exchange-correlation potential corresponding to a given second-order reduced density matrix. Unlike conventional Kohn-Sham inversion methods in which such potentials are extracted from ground-state electron densities, the proposed technique delivers unambiguous results in finite basis sets. The approach can also be used to separate approximately the exchange and correlation potentials for a many-electron system for which the reduced density matrix is known. The algorithm is implemented for configuration-interaction wave functions and its performance is illustrated with numerical examples. [ABSTRACT FROM AUTHOR]

One can partition the molecular density into its atomic contributions by minimizing the divergence of the atom-in-molecule densities from their corresponding reference pro-atomic densities, subject to the constraint that the sum of the atom-in-molecule densities is the total molecular density. We expose conditions on the divergence measure that are necessary, and sufficient, to recover the popular Hirshfeld partitioning. Specifically, among all local measures of the divergence between two probability distribution functions, the Hirshfeld partitioning is obtained only for f-divergences. [ABSTRACT FROM AUTHOR]

Identities for the difference of exchange-correlation potentials and energies in degenerate and nondegenerate ground states are derived. The constraints are strong for degenerate ground states, and suggest that local and semilocal approximations to the exchange-correlation energy functional are incapable of correctly treating degenerate ground states. For degenerate states, it is possible to provide both local (pointwise) equality and global inequality constraints for the exchange-correlation potential in terms of the Coulomb potential. [ABSTRACT FROM AUTHOR]

Density-functional-theory-based chemical reactivity indicators are formulated for degenerate and near-degenerate ground states. For degenerate states, the functional derivatives of the energy with respect to the external potential do not exist, and must be replaced by the weaker concept of functional variation. The resultant reactivity indicators depend on the specific perturbation. Because it is sometimes impractical to compute reactivity indicators for a specific perturbation, we consider two special cases: point-charge perturbations and Dirac delta function perturbations. The Dirac delta function perturbations provide upper bounds on the chemical reactivity. Reactivity indicators using the common used 'average of degenerate states approximation' for degenerate states provide a lower bound on the chemical reactivity. Unfortunately, this lower bound is often extremely weak. Approximate formulas for the reactivity indicators within the frontier-molecular-orbital approximation and special cases (two or three degenerate spatial orbitals) are presented in the supplementary material. One remarkable feature that arises in the frontier molecular orbital approximation, and presumably also in the exact theory, is that removing electrons sometimes causes the electron density to increase at the location of a negative (attractive) Dirac delta function perturbation. That is, the energetic response to a reduction in the external potential can increase even when the number of electrons decreases. [ABSTRACT FROM AUTHOR]

Algorithms for parallel unconstrained minimization of molecular systems are examined. The overall framework of minimization is the same except for the choice of directions for updating the quasi-Newton Hessian. Ideally these directions are chosen so the updated Hessian gives steps that are same as using the Newton method. Three approaches to determine the directions for updating are presented: the straightforward approach of simply cycling through the Cartesian unit vectors (finite difference), a concurrent set of minimizations, and the Lanczos method. We show the importance of using preconditioning and a multiple secant update in these approaches. For the Lanczos algorithm, an initial set of directions is required to start the method, and a number of possibilities are explored. To test the methods we used the standard 50-dimensional analytic Rosenbrock function. Results are also reported for the histidine dipeptide, the isoleucine tripeptide, and cyclic adenosine monophosphate. All of these systems show a significant speed-up with the number of processors up to about eight processors. [ABSTRACT FROM AUTHOR]

Three new algorithms are presented for determining transition state (TS) structures on the reduced potential energy surface, that is, for problems in which a few important degrees of freedom can be isolated. All three methods use constrained optimization to rapidly find the TS without an initial Hessian evaluation. The algorithms highlight how efficiently the TS can be located on a reduced surface, where the rest of the degrees of freedom are minimized. The first method uses a nonpositive definite quasi-Newton update for the reduced degrees of freedom. The second uses Shepard interpolation to fit the Hessian and starts from a set of points that bound the TS. The third directly uses a finite difference scheme to calculate the reduced degrees of freedom of the Hessian of the entire system, and searches for the TS on the full potential energy surface. All three methods are tested on an epoxide hydrolase cluster, and the ring formations of cyclohexane and cyclobutenone. The results indicate that all the methods are able to converge quite rapidly to the correct TS, but that the finite difference approach is the most efficient. [ABSTRACT FROM AUTHOR]

In this work we explore the physical foundations of models that study the variation of the ground state energy with respect to the number of electrons (E vs. N models), in terms of general grand-canonical (GC) ensemble formulations. In particular, we focus on E vs. N models that interpolate the energy between states with integer number of electrons. We show that if the interpolation of the energy corresponds to a GC ensemble, it is not differentiable. Conversely, if the interpolation is smooth, then it cannot be formulated as any GC ensemble. This proves that interpolation of electronic properties between integer electron numbers is inconsistent with any form of ensemble averaging. This emphasizes the role of derivative discontinuities and the critical role of a subsystem's surroundings in determining its properties. [ABSTRACT FROM AUTHOR]

The first purely density-based energy decomposition analysis (EDA) for intermolecular binding is developed within the density functional theory. The most important feature of this scheme is to variationally determine the frozen density energy, based on a constrained search formalism and implemented with the Wu–Yang algorithm [Q. Wu and W. Yang, J. Chem. Phys. 118, 2498 (2003)]. This variational process dispenses with the Heitler–London antisymmetrization of wave functions used in most previous methods and calculates the electrostatic and Pauli repulsion energies together without any distortion of the frozen density, an important fact that enables a clean separation of these two terms from the relaxation (i.e., polarization and charge transfer) terms. The new EDA also employs the constrained density functional theory approach [Q. Wu and T. Van Voorhis, Phys. Rev. A 72, 24502 (2005)] to separate out charge transfer effects. Because the charge transfer energy is based on the density flow in real space, it has a small basis set dependence. Applications of this decomposition to hydrogen bonding in the water dimer and the formamide dimer show that the frozen density energy dominates the binding in these systems, consistent with the noncovalent nature of the interactions. A more detailed examination reveals how the interplay of electrostatics and the Pauli repulsion determines the distance and angular dependence of these hydrogen bonds. [ABSTRACT FROM AUTHOR]

A new approach for computing the atom-in-molecule [quantum theory of atoms in molecule (QTAIM)] energies in Kohn–Sham density-functional theory is presented and tested by computing QTAIM energies for a set of representative molecules. In the new approach, the contribution for the correlation-kinetic energy (Tc) is computed using the density-functional theory virial relation. Based on our calculations, it is shown that the conventional approach where atomic energies are computed using only the noninteracting part of the kinetic energy might be in error by hundreds of kJ/mol. [ABSTRACT FROM AUTHOR]

Elements of Bader’s theory of atoms in molecules are combined with density-functional theory to provide an electron-preceding perspective on the deformation of materials. From this perspective, a network of atoms is changed by moving the bonds that connect them; the nuclei then follow. The electronic stress tensor is the key to understanding this process. Eigenvectors of the electronic stress tensor at critical points of the electron density provide insight into the “normal electronic modes” that accompany structural dynamics and rearrangements. Eigenvectors of the second-derivative matrix of the electron density emerge as effective approximations to the eigenvectors of the stress tensor; this makes it possible to apply our results to experimentally and computationally determined electron densities. To demonstrate the usefulness of our analysis, we show that (a) the low-frequency modes of ice Ic can be predicted from the eigenvectors of the second-derivative matrix and (b) the eigenvectors of the second-derivative matrix are associated with the direction of structural change during the pressure-induced phase transition from ice XI to a ferroelectric ice VIII-like structure. We conclude that the eigenvectors of the second-derivative matrix of the electron density are the key ingredient for constructing a dynamical theory of atoms in molecules. [ABSTRACT FROM AUTHOR]

Higher-order global softnesses, local softnesses, and softness kernels are defined along with their hardness inverses. The local hardness equalization principle recently derived by the authors is extended to arbitrary order. The resulting hierarchy of equalization principles indicates that the electronegativity/chemical potential, local hardness, and local hyperhardnesses all are constant when evaluated for the ground-state electron density. The new equalization principles can be used to test whether a trial electron density is an accurate approximation to the true ground-state density and to discover molecules with desired reactive properties, as encapsulated by their chemical reactivity indicators. [ABSTRACT FROM AUTHOR]

In the density-functional theory of chemical reactivity, the local hardness is known to be an ambiguous concept. The mathematical structure associated with this problematic situation is elaborated and three common definitions for the local hardness are critically examined: the frontier local hardness [S. K. Ghosh, Chem. Phys. Lett. 172, 77 (1990)], the total local hardness [S. K. Ghosh and M. Berkowitz, J. Chem. Phys. 83, 2976 (1985)], and the unconstrained local hardness [P. W. Ayers and R. G. Parr, J. Am. Chem. Soc. 122, 2010 (2000)]. The frontier local hardness has particularly nice properties: (a) it has smaller norm than most, if not all, other choices of the local hardness and (b) it is “unbiased” in an information-theoretic sense. For the ground electronic state of a molecular system, the frontier local hardness is equal to the global hardness. For an electronic system in its ground state, both the chemical potential and the frontier local hardness are equalized. The frontier local hardness equalization principle provides a computational approach for designing reagents with desirable chemical reactivity profiles. [ABSTRACT FROM AUTHOR]

By using perturbations in the molecular external potential, the authors deduce the Fukui function from the change in Kohn-Sham orbital energies, avoiding the troublesome differentiation of the density with respect to electron number. Though this paper focuses on the Fukui function, the same general technique can be used to compute the functional derivative of any observable with respect to the external potential. In this paper, the method is used to compute the Fukui function for the beryllium atom and the formaldehyde molecule. The follow-up paper (part II) addresses the problem of computing condensed reactivity indicators. [ABSTRACT FROM AUTHOR]

Stimulated by the difficulty of deriving effective kinetic energy functionals of the electron density, the authors consider using the local kinetic energy as the fundamental descriptor for molecular systems. In this ansatz, the electron density must be expressed as a functional of the local kinetic energy. There are similar results for other quantities, including the local temperature and the Kohn-Sham potential. One potential advantage of these approaches—and especially the approach based on the local temperature—is the chemical relevance of the fundamental descriptor. [ABSTRACT FROM AUTHOR]

The relaxation of the Pauli principle associated with density scaling is examined. Scaling the density has been investigated in the development of density functional computational methods with higher accuracy. Scaling the density by ρ(r)ζ=ρ(r)/ζ reduces the number of electrons to M=N/ζ when ζ>1. The minimum kinetic energy of the scaled density, Ts[ρ/ζ], can be scaled back to the N-electron system by multiplying the M-electron Kohn-Sham-type occupation numbers by ζ to produce Tζ[ρ]. This relaxes the Pauli principle when the orbital occupation numbers are greater than 1 in the N-electron system. The effects of antisymmetry on solutions to the Kohn-Sham equations are examined for Ne and the Be isoelectronic series. The changes in Tζ[ρ] and the exchange energy Exζ[ρ] when ζ is varied show that these two quantities are inextricably linked. [ABSTRACT FROM AUTHOR]

The Woodward-Hoffmann rules for pericyclic reactions, a fundamental set of reactivity rules in organic chemistry, are formulated in the language of conceptual density functional theory (DFT). DFT provides an elegant framework to introduce chemical concepts and principles in a quantitative manner, partly because it is formulated without explicit reference to a wave function, on whose symmetry properties the Woodward-Hoffmann [J. Am. Chem. Soc. 87, 395 (1965)] rules are based. We have studied the initial chemical hardness response using a model reaction profile for two prototypical pericyclic reactions, the Diels-Alder cycloaddition of 1,3-butadiene to ethylene and the addition of ethylene to ethylene, both in the singlet ground state and in the first triplet excited state. For the reaction that is thermally allowed but photochemically forbidden, the initial hardness response is positive along the singlet reaction profile. (By contrast, for the triplet reaction profile, a negative hardness response is observed.) For the photochemically allowed, thermally forbidden reaction, the behavior of the chemical hardness along the initial stages of the singlet and triplet reaction profiles is reversed. This constitutes a first step in showing that chemical concepts from DFT can be invoked to explain results that would otherwise require invoking the phase of the wave function. [ABSTRACT FROM AUTHOR]

We provide a rigorous proof that the Hohenberg-Kohn theorem holds for spin densities by extending Lieb’s Legendre-transform formulation to spin densities. The resulting spin-density-functional theory resolves several troublesome issues. Most importantly, the present paper provides an explicit construction for the spin potentials at any point along the adiabatic connection curve, thus providing a formal basis for the use of exchange-correlation functionals of the spin density in the Kohn-Sham density-functional theory (DFT). The practical implications of this result for unrestricted Kohn-Sham DFT calculations is considered, and the existence of holes below the Fermi level is discussed. We argue that an orbital’s energy tends to increase as its occupation number increases, which provides the basis for a computational algorithm for determining the occupation numbers in Kohn-Sham DFT and helps explain the origin of Hund’s rules and holes below the Fermi level. [ABSTRACT FROM AUTHOR]

A comprehensive analysis is presented for the acid-base double-exchange reaction as well as the associated acid-displacement and base-displacement “half-reactions” with the goal of elucidating the meaning of the hard/soft acid/base (HSAB) principle and the conditions for its validity. When electron-transfer effects are important and other effects are negligible, the HSAB principle is driven by the surpassing stability of the soft acid/soft base product. When electrostatic effects dominate the reactivity, the HSAB principle is driven by the surpassing stability of the hard acid/hard base product. Because electron-transfer effects favor soft/soft interactions, while electrostatic effects favor hard/hard interactions, acid-base exchange reactions may be used to determine whether a reagent’s reactivity is dominated by electron-transfer or by electrostatic effects. Because electron-transfer and electrostatic considerations separately favor the HSAB principle whenever the electronic chemical potentials of the acids and bases involved in the reaction are similar, our analysis provides strong support for the HSAB principle. The electronic chemical potential measures the intrinsic strength of acids and bases. [ABSTRACT FROM AUTHOR]

Given a general, N-particle Hamiltonian operator, analogs of the Hohenberg-Kohn theorem are derived for functions that are more general than the particle density, including density matrices and the diagonal elements thereof. The generalization of Lieb’s Legendre transform ansatz to the generalized Hohenberg-Kohn functional not only solves the υ-representability problem for these entities, but, more importantly, also solves the N-representability problem. Restricting the range of operators explored by the Legendre transform leads to a lower bound on the true functional. If all the operators of interest are incorporated in the restricted maximization, however, the variational principle dictates that exact results are obtained for the systems of interest. This might have important implications for practical work not only for density matrices but also for density functionals. A follow-up paper will present a useful alternative approach to the v- and N-representability problems based on the constrained search formalism. [ABSTRACT FROM AUTHOR]

A variational method for forcing the exchange-correlation potential in density-functional theory (DFT) to have the correct asymptotic decay is developed. The resulting exchange-correlation potentials are much improved while the total energies remain essentially the same, compared with conventional density-functional theory calculations. The highest occupied orbital energies from the asymptotically corrected exchange-correlation potentials are found to provide significantly more accurate approximations to the ionization potential (for a neutral molecule) and the electron affinity (for an anion) than those from conventional calculations, although the results are usually inferior to direct methods by computing energy differences. Extending recent results from exchange-only DFT, we show that exact exchange-correlation potential is nonuniform asymptotically. Correcting the asymptotic decay of approximate exchange-correlation potentials towards the exact functional form binds the highest occupied orbitals for atomic and molecular anions, which supports the use of DFT calculations for negatively charged molecular species. With this technique, even hybrid functionals have local exchange-correlation potentials, effectively removing the largest objection to including these functionals in the panoply of Kohn-Sham DFT methods. [ABSTRACT FROM AUTHOR]

Two recent papers [P. W. Ayers and R. G. Parr, J. Am. Chem. Soc. 122, 2010 (2000); 123, 2007 (2001)] have shown how variational principles for the energy may be used to derive and elucidate the significance of the chemical reactivity indices of density-functional theory. Here, similar ideas are applied, yielding a systematic, mathematically rigorous, and physically sound approach to condensed reactivity indices. First, we use the variational principle for the energy to derive an expression for the condensed Fukui function index in terms of the condensed hardness kernel. Next, we address an important open problem pertaining to condensed reactivity indices: when (if ever) is the condensed Fukui function for an atom in a molecule negative? In particular, our analysis confirms the observation, hitherto based only on computational evidence, that the Hirshfeld partitioning is optimal for obtaining non-negative Fukui functions. We also hypothesize that the strong diagonal dominance of the condensed hardness kernel is sufficient for the non-negativity of the Fukui function. Errors in the partitioning of molecules into atoms and inadequate treatment of correlation are pinpointed as the most likely causes of negative condensed Fukui functions. We conclude by noting that the condensed Fukui functions are, in some respects, more appropriate indicators of a molecular site’s reactivity than the Fukui function itself. © 2002 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Examines the axiomatic basis for the concept of an atom within a molecule. Difficulty in providing a satisfactory mathematical basis; Definition of transferability; Description of Hirshfeld atom in a molecule.

The atoms in molecules associated with the Hirshfeld partitioning minimize the generalized Hellinger-Bhattacharya distance to the reference pro-atom densities. Moreover, the reference proatoms can be chosen by minimizing the distance between the pro-molecule density and the true molecular density. This provides an alternative to both the heuristic "stockholder" and the mathematical information-theoretic interpretations of the Hirshfeld partitioning. These results extend to any member of the family of f-divergences. [ABSTRACT FROM AUTHOR]

Grid is a free and open-source Python library for constructing numerical grids to integrate, interpolate, and differentiate functions (e.g., molecular properties), with a strong emphasis on facilitating these operations in computational chemistry and conceptual density functional theory. Although designed, maintained, and released as a stand-alone Python library, Grid was originally developed for molecular integration, interpolation, and solving the Poisson equation in the HORTON and ChemTools packages. Grid is designed to be easy to use, extend, and maintain; this is why we use Python and adopt many principles of modern software development, including comprehensive documentation, extensive testing, continuous integration/delivery protocols, and package management. We leverage popular scientific packages, such as NumPy and SciPy, to ensure high efficiency and optimized performance in grid development. This article is the official release note of the Grid library showcasing its unique functionality and scope. [ABSTRACT FROM AUTHOR]

Discusses the alternative definition of exchange-correlation charge in density functional theory (DFT). Numerical evaluation of the exchange-correlation charge by using the Zhao-Morrison-Parr method; Radial distribution of exchange-correlation charge and atomic density for atom neon; Comparison of accurate exchange-correlation charge for atom argon.

Approximate generalized overlap amplitudes (GOAs), also called Feynman–Dyson amplitudes, have been calculated from a full configuration interaction (CI) wave function for the ground state of beryllium using the extended Koopmans’ theorem (EKT). The GOAs were also calculated from the ground state CI wave function for Be and the ground state and excited state CI wave functions for Be+. The EKT GOAs are nearly identical to the corresponding CI GOAs for the lower 2S states of Be+ and for the Be+ 1s2s2 state which has a large GOA occupation number. There are many CI GOAs for which there is not a corresponding EKT GOA. This may be due in part to the limited size of the basis set and to the inability to include natural spin orbitals with small occupation numbers in the EKT calculations. © 1995 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We show that the hard/soft acid/base principle holds when electron-transfer effects are dominant and the weaker acid and stronger base are harder than the other acidic and basic reagents. In this case the preference of strong acids for strong bases and weak acids for weak bases reinforces the preference of hard acids for hard bases and soft acids for soft bases. [ABSTRACT FROM AUTHOR]

The hard/soft-acid/base (HSAB) principle indicates that hard acids prefer binding to hard bases (often forming bonds with substantial ionic character) while soft acids prefer binding to soft bases (often forming bonds with substantial covalent character). Though the HSAB principle is a foundational concept of the modern theory of acids and bases, the theoretical underpinnings of the HSAB principle remain murky. This paper examines the exchange reaction, wherein two molecules, one the product of reacting a hard acid and a soft base and the other the product of reacting a soft acid with a hard base, exchange substituents to form the preferred hard–hard and soft–soft product. A simple derivation shows that this reaction is exothermic, proving the validity of the HSAB principle. The analysis leads to the simple and conceptually appealing conclusion that the HSAB principle is a driven by simple electron transfer effects. [ABSTRACT FROM AUTHOR]

We provide new arguments supporting the validity of the maximum hardness and the minimum electrophilicity principles, considering the overall change of these descriptors in a charge-transfer reaction. We analyze two cases: (a) how the reactivity is affected when we perturb one reagent, keeping the other constant; (b) how the hardness and electrophilicity change when we treat the interaction between the reagents as a perturbation. [ABSTRACT FROM AUTHOR]

HORTON is a free and open-source electronic-structure package written primarily in Python 3 with some underlying C++ components. While HORTON's development has been mainly directed by the research interests of its leading contributing groups, it is designed to be easily modified, extended, and used by other developers of quantum chemistry methods or post-processing techniques. Most importantly, HORTON adheres to modern principles of software development, including modularity, readability, flexibility, comprehensive documentation, automatic testing, version control, and quality-assurance protocols. This article explains how the principles and structure of HORTON have evolved since we started developing it more than a decade ago. We review the features and functionality of the latest HORTON release (version 2.3) and discuss how HORTON is evolving to support electronic structure theory research for the next decade. [ABSTRACT FROM AUTHOR]

Electron pairs have an illustrious history in chemistry, from powerful concepts to understanding structural stability and reactive changes to the promise of serving as building blocks of quantitative descriptions of the electronic structure of complex molecules and materials. However, traditionally, two-electron wavefunctions (geminals) have not enjoyed the popularity and widespread use of the more standard single-particle methods. This has changed recently, with a renewed interest in the development of geminal wavefunctions as an alternative to describing strongly correlated phenomena. Hence, there is a need to find geminal methods that are accurate, computationally tractable, and do not demand significant input from the user (particularly via cumbersome and often ill-behaved orbital optimization steps). Here, we propose new families of geminal wavefunctions inspired by the pair coupled cluster doubles ansatz. We present a new hierarchy of two-electron wavefunctions that extends the one-reference orbital idea to other geminals. Moreover, we show how to incorporate single-like excitations in this framework without leaving the quasiparticle picture. We explore the role of imposing seniority restrictions on these wavefunctions and benchmark these new methods on model strongly correlated systems. [ABSTRACT FROM AUTHOR]

In methods like geminal-based approaches or coupled cluster that are solved using the projected Schrödinger equation, direct computation of the 2-electron reduced density matrix (2-RDM) is impractical and one falls back to a 2-RDM based on response theory. However, the 2-RDMs from response theory are not N-representable. That is, the response 2-RDM does not correspond to an actual physical N-electron wave function. We present a new algorithm for making these non-N-representable 2-RDMs approximately N-representable, i.e., it has the right symmetry and normalization and it fulfills the P-, Q-, and G-conditions. Next to an algorithm which can be applied to any 2-RDM, we have also developed a 2-RDM optimization procedure specifically for seniority-zero 2-RDMs. We aim to find the 2-RDM with the right properties which is the closest (in the sense of the Frobenius norm) to the non-N-representable 2-RDM by minimizing the square norm of the difference between this initial response 2-RDM and the targeted 2-RDM under the constraint that the trace is normalized and the 2-RDM, Q-matrix, and G-matrix are positive semidefinite, i.e., their eigenvalues are non-negative. Our method is suitable for fixing non-N-representable 2-RDMs which are close to being N-representable. Through the N-representability optimization algorithm we add a small correction to the initial 2-RDM such that it fulfills the most important N-representability conditions. [ABSTRACT FROM AUTHOR]

We have witnessed considerable research interest in the recent literature about the development and applications of quantities from the information-theoretic approach (ITA) in density functional theory. These ITA quantities are explicit density functionals, whose local distributions in real space are continuous and well-behaved. In this work, we further develop ITA by systematically analyzing the topological behavior of its four representative quantities, Shannon entropy, two forms of Fisher information, and relative Shannon entropy (also called information gain or Kullback−Leibler divergence). Our results from their topological analyses for 103 molecular systems provide new insights into bonding interactions and physiochemical properties, such as electrophilicity, nucleophilicity, acidity, and aromaticity. We also compare our results with those from the electron density, electron localization function, localized orbital locator, and Laplacian functions. Our results offer a new methodological approach and practical tool for applications that are especially promising for elucidating chemical bonding and reactivity propensity. [ABSTRACT FROM AUTHOR]

A recent paper [P. W. Ayers, J. Chem. Phys122, 141102 (2005)] considered the hard/soft acid/base exchange reaction, showing that the products associated with the hard/soft acid/base rule (in which the hard acid and hard base are bound, as are the soft acid and soft base) have lower energy than the alternative (in which the hard acid and soft base would have been bound and similarly the soft acid and hard base). Here we show that the maximum hardness principle also predicts this result. Unlike the previous derivation, we do not need to make any assumptions about the relative strength of the acids and bases. [ABSTRACT FROM AUTHOR]

Condensed-to-atom Fukui functions which reflect the atomic reactivity like the tendency susceptible to either nucleophilic or electrophilic attack demonstrate the bonding trend of an atom in a molecule. Accordingly, Fukui functions based concepts, that is, bonding reactivity descriptors which reveal the bonding properties of molecules in the reaction were put forward and then applied to pericyclic and cluster reactions to confirm their effectiveness and reliability. In terms of the results from the bonding descriptors, a covalent bond can readily be predicted between two atoms with large Fukui functions (i.e., one governs nucleophilic attack while the other one governs electrophilic attack, or both of them govern radical attacks) for pericyclic reactions. For SinOm clusters' reactions, the clusters with a low O atom ratio readily form a bond between two Si atoms with big values of their Fukui functions in which they respectively govern nucleophilic and electrophilic attacks or both govern radical attacks. Also, our results from bonding descriptors show that Si—Si bonds can be formed via the radical mechanism between two Si atoms, and formations of Si—O and O—O bonds are possible when the O content is high. These results conform with experimental findings and can help experimentalists design appropriate clusters to synthesize Si nanowires with high yields. The approach established in this work could be generalized and applied to study reactivity properties for other systems. [ABSTRACT FROM AUTHOR]

We analyze the minimum electrophilicity principle of conceptual density functional theory using the framework of the finite temperature grand canonical ensemble. We provide support for this principle, both for the cases of systems evolving from a non-equilibrium to an equilibrium state and for the change from one equilibrium state to another. In doing so, we clearly delineate the cases where this principle can, or cannot, be used. [ABSTRACT FROM AUTHOR]

An alternative definition of hardness (called the thermodynamic hardness) within the grand canonical ensemble formalism is proposed in terms of the partial derivative of the electronic chemical potential with respect to the thermodynamic chemical potential of the reservoir, keeping the temperature and the external potential constant. This temperature dependent definition may be interpreted as a measure of the propensity of a system to go through a charge transfer process when it interacts with other species, and thus it keeps the philosophy of the original definition. When the derivative is expressed in terms of the three-state ensemble model, in the regime of low temperatures and up to temperatures of chemical interest, one finds that for zero fractional charge, the thermodynamic hardness is proportional to T-1(I - A), where I is the first ionization potential, A is the electron affinity, and T is the temperature. However, the thermodynamic hardness is nearly zero when the fractional charge is different from zero. Thus, through the present definition, one avoids the presence of the Dirac delta function. We show that the chemical hardness defined in this way provides meaningful and discernible information about the hardness properties of a chemical species exhibiting integer or a fractional average number of electrons, and this analysis allowed us to establish a link between the maximum possible value of the hardness here defined, with the minimum softness principle, showing that both principles are related to minimum fractional charge and maximum stability conditions. [ABSTRACT FROM AUTHOR]

Two recent papers [S. Liu, P. W. Ayers, and R. G. Parr, J. Chem. Phys. 111, 6927 (1999); A. Go¨rling, Phys. Rev. Lett. 83, 5459 (1999)] have stated that ∫∇[sup 2]ν[sub xc](r)dr=4π, where ν[sub xc](r) is the exchange-correlation potential of density functional theory. Here, we derive this sum rule and related rules such as ∫∇[sup 2]ν[sub x](r)dr=4π and ∫∇[sup 2]ν[sub c](r)dr=0, where ν[sub x](r) and ν[sub c](r) are the exchange and correlation components of ν[sub xc](r). Using similar methods, we derive the sum rule for the “screening” portion of the exchange-correlation potential and also “generalized” sum rules for ν[sub c](r) and the “response” portion of the exchange-correlation potential, v[sub xc][sup response](r). From the sum rule for v[sub xc][sup response](r), we deduce the asymptotic decay of the density response of the hole-correlation function. We conclude by discussing the probable utility of these results for the development of new exchange-correlation functionals. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We extend the definition of the electronic chemical potential (µe) and chemical hardness (e) to finite temperatures by considering a reactive chemical species as a true open system to the exchange of electrons, working exclusively within the framework of the grand canonical ensemble. As in the zero temperature derivation of these descriptors, the response of a chemical reagent to electron-transfer is determined by the response of the (average) electronic energy of the system, and not by intrinsic thermodynamic properties like the chemical potential of the electron-reservoir which is, in general, different from the electronic chemical potential, μe. Although the dependence of the electronic energy on electron number qualitatively resembles the piecewise-continuous straight-line profile for low electronic temperatures (up to ca. 5000 K), the introduction of the temperature as a free variable smoothens this profile, so that derivatives (of all orders) of the average electronic energy with respect to the average electron number exist and can be evaluated analytically. Assuming a three-state ensemble, well-known results for the electronic chemical potential at negative (-I), positive (-A), and zero values of the fractional charge (-(I + A)/2) are recovered. Similarly, in the zero temperature limit, the chemical hardness is formally expressed as a Dirac delta function in the particle number and satisfies the well-known reciprocity relation with the global softness. [ABSTRACT FROM AUTHOR]

We develop a variational procedure for the iterative Hirshfeld (HI) partitioning scheme. The main practical advantage of having a variational framework is that it provides a formal and straightforward approach for imposing constraints (e.g., fixed charges on certain atoms or molecular fragments) when computing HI atoms and their properties. Unlike many other variants of the Hirshfeld partitioning scheme, HI charges do not arise naturally from the information-theoretic framework, but only as a reverse-engineered construction of the objective function. However, the procedure we use is quite general and could be applied to other problems as well. We also prove that there is always at least one solution to the HI equations, but we could not prove that its self-consistent equations would always converge for any given initial pro-atom charges. Our numerical assessment of the constrained iterative Hirshfeld method shows that it satisfies many desirable traits of atoms in molecules and has the potential to surpass existing approaches for adding constraints when computing atomic properties. [ABSTRACT FROM AUTHOR]

We present an improved electronic linear response model to incorporate polarization and charge-transfer effects in polarizable force fields. This model is a generalization of the Atom-Condensed Kohn-Sham Density Functional Theory (DFT), approximated to second order (ACKS2): it can now be defined with any underlying variational theory (next to KS-DFT) and it can include atomic multipoles and off-center basis functions. Parameters in this model are computed efficiently as expectation values of an electronic wavefunction, obviating the need for their calibration, regularization, and manual tuning. In the limit of a complete density and potential basis set in the ACKS2 model, the linear response properties of the underlying theory for a given molecular geometry are reproduced exactly. A numerical validation with a test set of 110 molecules shows that very accurate models can already be obtained with fluctuating charges and dipoles. These features greatly facilitate the development of polarizable force fields. [ABSTRACT FROM AUTHOR]

In analogy to the electrophilicity, we define potentialphilicity indicators that represent energetically favorable ways to change the external potential of a molecule at fixed electron number. Similarly, we define a potentialphobicity to represent the least favorable way to change the external potential of a molecule. The resulting indicators should be useful for describing how molecular geometries change and predicting favorable and unfavorable ways for a reagent to approach a molecule. The linear response function enters plays a very important role in this approach, analogous to the role of the hardness for the electrophilicity or the hardness kernel for the Fukui function. The mathematical properties of the response function and its implications for these reactivity indicators are discussed in depth. [ABSTRACT FROM AUTHOR]

A simple explanation is given for the exactness of the extended Koopmans’ theorem, (EKT) for computing the removal energy of any many-electron system to the lowest-energy ground state ion of a given symmetry. In particular, by removing the electron from a “removal orbital” of appropriate symmetry that is concentrated in the asymptotic region, one obtains the exact ionization potential and the exact Dyson orbital for the corresponding state of the ion. It is argued that the EKT is not restricted to many-electron systems but holds for any finite many-body system, provided that the interaction vanishes for increasing interparticle distance. A necessary and sufficient condition for the validity of the EKT for any state (not just the lowest-energy states of a given symmetry) in terms of the third-order reduced density matrix is stated and derived. [ABSTRACT FROM AUTHOR]

A new numerical integration procedure for exchange-correlation energies and potentials is proposed and “proof of principle” results are presented. The numerical integration grids are built from sparse-tensor product grids (constructed according to Smolyak’s prescription [Dokl. Akad. Nauk. 4, 240 (1963)] ) on the unit cube. The grid on the unit cube is then transformed to a grid over real space with respect to a weight function, which we choose to be the promolecular density. This produces a “whole molecule” grid, in contrast to conventional integration methods in density-functional theory, which use atom-in-molecule grids. The integration scheme was implemented in a modified version of the DEMON2K density-functional theory program, where it is used to evaluate integrals of the exchange-correlation energy density and the exchange-correlation potential. Ground-state energies and molecular geometries are accurately computed. The biggest advantages of the grid are its flexibility (it is easy to change the number and distribution of grid points) and its whole molecule nature. The latter feature is potentially helpful for basis-set-free computational algorithms. [ABSTRACT FROM AUTHOR]

This paper supersedes previous theoretical approaches to conceptual DFT because it provides a unified and systematic approach to all of the commonly considered formulations of conceptual DFT, and even provides the essential mathematical framework for new formulations. Global, local, and nonlocal chemical reactivity indicators associated with the “closed-system representation” ([Nα,Nβ,να(r),νβ(r)]) of spin-polarized density functional theory (SP-DFT) are derived. The links between these indicators and the ones associated with the “open-system representation” ([μα,μβ,να(r),νβ(r)]) are derived, including the spin-resolved Berkowitz–Parr identity. The Legendre transform to the “density representation” ([ρα(r),ρβ(r)]) is performed, and the spin-resolved Harbola–Chattaraj–Cedillo–Parr identities linking the density representation to the closed-system and open-system representations are derived. Taken together, these results provide the framework for understanding chemical reactions from both the electron-following perspective (using either the closed-system or the open-system representation) and electron-preceding perspective (density representation). A powerful matrix-vector notation is developed; with this notation, identities in conceptual DFT become universal. Specifically, this notation allows the fundamental identities in conventional (spin-free) conceptual DFT, the [Nα,Nβ] representation, and the [N=Nα+Nβ,NS=Nα-Nβ] representation to be written in exactly the same forms. In cases where spin transfer and electron transfer are coupled (e.g., radical+molecule reactions), we believe that the [Nα,Nβ] representation may be more useful than the more common [N,NS] representation. [ABSTRACT FROM AUTHOR]

The Fukui function is a frequently used DFT concept in the description of a system’s regioselective preferences to undergo electrophilic, nucleophilic, or radical attacks. Until now, this function has usually been evaluated using finite difference approximations. The first paper in this series proposed a method for obtaining the Fukui function by a direct calculation of the functional derivative of the chemical potential with respect to the external potential. This paper extends the method to condensed Fukui functions and applies it to an extensive testing set of molecules. Results are promising, which demonstrates the usefulness of the new formalism. [ABSTRACT FROM AUTHOR]